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Second-order PDEs in four dimensions with half-flat conformal structure

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posted on 06.01.2020, 15:36 by Sobhi BerjawiSobhi Berjawi, Evgeny FerapontovEvgeny Ferapontov, Boris Kruglikov, Vladimir NovikovVladimir Novikov
We study second-order partial differential equations (PDEs) in four dimensions for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge–Ampère property. Since half-flatness of the conformal structure is equivalent to the existence of a non-trivial dispersionless Lax pair, our result explains the observation that all known scalar second-order integrable dispersionless PDEs in dimensions four and higher are of Monge–Ampère type. Some partial classification results of Monge–Ampère equations in four dimensions with half-flat conformal structure are also obtained.

Funding

EPSRC grant EP/N031369/1

Trond Mohn Foundation

Tromsø Research Foundation

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volume

476

Issue

2233

Publisher

The Royal Society

Version

AM (Accepted Manuscript)

Rights holder

© The Authors

Publisher statement

This paper : Berjawi S, Ferapontov EV, Kruglikov B, Novikov V. 2020 Second-order PDEs in four dimensions with half-flat conformal structure. Proc. R. Soc. A 476: 20190642 was accepted for publication in the journal Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences and the definitive published version is available at https://doi.org/10.1098/rspa.2019.0642.

Acceptance date

18/12/2019

Publication date

2020-01-29

Copyright date

2020

ISSN

1364-5021

eISSN

1471-2946

Language

en

Depositor

Prof Evgeny Ferapontov. Deposit date: 29 December 2019

Article number

20190642